One considers the class of maps u:D → S2, which map the boundary of D to one point in S2. If u
were also harmonic, then it is known that u must be constant. However, if u is instead
f-harmonic -- a critical point of the energy functional 1/2 ∫D f(x) |∇ u(x)|2 --
then this need not be true. We shall see that there exist functions f:D → (0,∞) and
nonconstant f-harmonic maps u:D → S2 which map the boundary to one point. We will also
see that there exist nonconstant f for which, there is no nonconstant f-harmonic map in this
class. Finally, we see that there exists a nonconstant f-harmonic map from the torus to the 2-sphere.

Acknowledgements

Research supported by Swiss National Science Foundation grant number 200020-107652/1 and EPSRC award number 00801877.